Thomas J. Haines

Nearby Cycles for Local Models of Some Shimura Varieties

Kottwitz conjectured a formula for the (semi-simple) trace of Frobenius on the nearby cycles for the local model of a Shimura variety with Iwahori-type level structure. In this paper, we prove his conjecture in the linear and symplectic cases by adapting an argument of Gaitsgory, who proved an analogous theorem in the equal characteristic case.

The combinatorics of Bernstein functions

A construction of Bernstein associates to each cocharacter of a split p-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz’ conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of Gln; one example can be used to verify Kottwitz’ conjecture for a special class of Shimura varieties (the “Drinfeld case”). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the “μ-admissible” set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.

AFFINE DELIGNE-LUSZTIG VARIETIES IN AFFINE FLAG VARIETIES

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic

The Satake isomorphism for special maximal parahoric Hecke algebras

\sigma

Test functions for Shimura varieties: the Drinfeld case

-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic

The base change fundamental lemma for central elements in parahoric Hecke algebras

\sigma

The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties

-conjugacy class is the class of

Structure constants for Hecke and representation rings

b=1

Frobenius semisimplicity for convolution morphisms

, we also prove the dimension formula predicted in op. cit. in almost all cases.

Introduction to Shimura Varieties with Bad Reduction of Parahoric Type

Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. As an application we define a transfer homomorphism t : HK∗ (G∗) → HK(G) where G∗ is the quasi-split inner form of G. We also describe how our results relate to the treatment of Cartier [Car], where K is replaced by a special maximal compact open subgroup K ⊂ G(F ) and where a Satake isomorphism is established for the Hecke algebra H K .